Complete Defectivity and Simplex-Range Rational Identifiability for Homoscedastic Gaussian Moment Varieties

Abstract

This work gives a complete dimension and defectivity classification for homoscedastic Gaussian moment varieties over an algebraically closed field of characteristic zero. It proves that these varieties have the expected dimension for every moment order at least four and combines this result with the known cubic classification to determine all defective cases. The paper also establishes a uniform nondefectivity theorem when the common covariance is restricted to a general positive-dimensional linear subspace, including the isotropic covariance model. In addition, it determines rational identifiability throughout the simplex range. The least rationally identifying moment order is two for one component, five for two components, and four for every model with at least three components in the simplex range. The proofs combine a degeneration of the common covariance tangent block, fat-point postulation, Waring decomposition methods, cumulant coordinates, and flat moment matrices. The three- and four-component cases are treated separately by explicit rational reconstruction and an exact reduced Gröbner-fiber argument with boundary exclusion. The accompanying computation archive provides executable exact-arithmetic verification code, fixed inputs, deterministic outputs, and a unified reproduction command. It includes independent checks of Jacobian ranks, the cubic defect classification, quartic recovery, restricted covariance models, the numerical conditions entering the fat-point argument, and the exact small-component fiber certificates. Research methodology and AI assistance:This work was developed using the CARMA-Math research workflow, a cumulative AI-assisted mathematical research methodology using persistent research archives, literature and prior-art investigation, iterative proof exploration, and verification procedures. Generative AI (ChatGPT) was used extensively for mathematical exploration, proof development, computational reasoning, literature research, and manuscript preparation.